Determine (dy/dx) using implicit differentiation.

cos(X^2Y^2) = x

I'm really confused what to do now..i think the next steps are:

d/dx [cos(X^2*Y^2)] = d/dx [x]
= -sin(X^2*Y^2)* ((X^2*2Y dy/dx) + (Y^2*2X)) = 1
= -2YX^2 sin(X^2*Y^2) dy/dx + -2XY^2sin(X^2*Y^2) = 1
= -2YX^2 sin(X^2*Y^2) dy/dx = 1 + -2XY^2sin(X^2*Y^2)
= dy/dx = (1 + -2XY^2sin(X^2*Y^2))/ (-2YX^2 sin(X^2*Y^2))

Can someone tell me if this is correct?

Yes, your steps are correct! You have correctly applied implicit differentiation to find the derivative dy/dx.

To find dy/dx, you started by taking the derivative of both sides of the equation with respect to x. Then, you used the chain rule and the product rule appropriately.

Your final result for dy/dx is:

dy/dx = (1 + -2XY^2sin(X^2*Y^2))/ (-2YX^2 sin(X^2*Y^2))

Well done!

Yes, your steps for finding the derivative (dy/dx) using implicit differentiation are correct.

To solve the equation cos(X^2Y^2) = x for the derivative (dy/dx), you differentiate both sides of the equation with respect to x.

Starting from d/dx [cos(X^2Y^2)] = d/dx [x], you correctly apply the chain rule to differentiate the left side. The derivative of cos(u) is -sin(u) times the derivative of u with respect to x. In this case, u is X^2Y^2, so the derivative of cos(X^2Y^2) with respect to x is -sin(X^2Y^2) multiplied by the derivative of X^2Y^2 with respect to x.

You correctly find the derivative of X^2Y^2 with respect to x by using the product rule: (X^2*2Y dy/dx) + (Y^2*2X).

Then, you set the derivative of cos(X^2Y^2) equal to 1, as both sides of the equation are equal.

Next, you rearrange the equation and isolate dy/dx. You move the terms without dy/dx to the other side of the equation and divide both sides by -2YX^2sin(X^2Y^2) to solve for dy/dx.

So, your final answer is dy/dx = (1 + -2XY^2sin(X^2Y^2))/ (-2YX^2 sin(X^2Y^2)). Well done!